Sometimes collective information reveals something that no individual response could confirm.
The natives in a circle
Riddle statement
An anthropologist is surrounded by a circle of natives.
Each one belongs to one of two tribes:
- those who always tell the truth;
- those who always lie.
Each native is asked the same question:
“Does the person on your right always tell the truth?”
After listening to all the answers, the anthropologist still does not know who belongs to which tribe.
But she does discover something surprising: the proportion of liars is determined in a unique way.
What is that proportion?
Show solution
Solution
Let's call V a truthful person and M a liar.
Let's analyze what each native answers about their right neighbor:
A truthful with a neighbor tells the truth): answers no.
A liar with neighbor M lies (the neighbor does not tell the truth): answers yes.
In summary: any native answers yes if his right neighbor is from the same tribe, and no if he is from a different tribe.
The answers, therefore, only reveal whether each pair of consecutive neighbors matches in type or not. They do not allow us to know who is V and who is M in absolute terms.
This implies that, given any distribution compatible with the answers, there is always another equally valid one: it is enough to exchange all the Vs for M and all the Ms for V. The answers of the circle do not change at all.
However, the statement establishes that the proportion of liars is determined in a unique way. The only way for a proportion to be invariant when exchanging truth-tellers and liars is for both groups to be exactly the same size.
Therefore, half of the natives lie and the other half tell the truth.